AP Statistics

Issues Interpreting Correlation and Regression  Limitations for r, r 2, and LSRL :  Can only be used to describe linear relationships.  r, r 2, and LSRL can be influenced by extremes.  Extrapolation: Predicting outside the domain of x. Such predictions can’t be trusted.  Lurking Variables: Sometimes a relationship between x and y can be influenced by other variables without our knowledge.  Using Averaged Data (combining info from many individuals): These correlations are usually too high as the usual variation from individuals is lost.

Why Associations Happen  Causation: Changes in x cause changes in y.  Example: A drop in temperature causes increased power consumption.

Why Associations Happen  Common Response: Both x and y respond to changes in some observed variable or variables.

There is an almost perfect linear relationship between x and y. (r= ) x = # Methodist Ministers in New England y = # of Barrels of Rum Imported to Boston CORRELATION DOES NOT IMPLY CAUSATION! x = # Methodist Ministers in New England y = # of Barrels of Rum Imported to Boston CORRELATION DOES NOT IMPLY CAUSATION! Example of Common Response

 Methodist Ministers vs Barrels of Rum Imported to Boston.  r =  What is the variable driving the growth in the number of ministers and the number of barrels of rum being imported?

Why Associations Happen  Confounding: The effect on y of the explanatory variable x is hopelessly mixed up with the effects on y by other variables.

Example of Confounding  We think that the amount of calories consumed each day is related a person’s body mass index.  Body mass index may be based on genetics, too.

Association doesn’t imply causation!  An association between x and y, even if strong, is not good evidence that changes in x cause changes in y.  The best way to get evidence that x causes y is to do an experiment.  Experiments control lurking variables, so that there isn’t any other explanation for y changing as x changes.

Confounding vs. Lurking Variables  There are any number of variables in the environment during a study or an experiment. These are lurking variables.  Once any one of these variables add unwanted variation to our results, they become confounding variables.

Examples 1) There is a high positive correlation: nations with many TV sets have higher life expectancies. Could we lengthen the life of people in Rwanda by shipping them TVs? 2) People who use artificial sweeteners in place of sugar tend to be heavier than people who use sugar. Does artificial sweetener use cause weight gain? 3) Women who work in the production of computer chips have abnormally high numbers of miscarriages. The union claimed chemicals cause the miscarriages. Another explanation may be the fact these workers spend a lot of time on their feet.

4) People with two cars tend to live longer than people who own only one car. Owning three cars is even better, and so on. What might explain the association? 5) Children who watch many hours of TV get lower grades on average than those who watch less TV. Why does this fact not show that watching TV causes low grades?

6) Data show that married men (and men who are divorced or widowed) earn more than men who have never been married. If you want to make more money, should you get married? 7) High school students who take the SAT, enroll in an SAT coaching course, and take the SAT again raise their mathematics score from an average of 521 to 561. Can this increase be attributed entirely to taking the course?

Homework  Textbook: 4.33, 4.37, 4.38, 4.42, 4.44, 4.76